System and method for modeling affinity and cannibalization in customer buying decisions

ABSTRACT

A computer system models customer response using observable data. The observable data includes transaction, product, price, and promotion. The computer system receives data observable from customer responses. A set of factors including customer traffic within a store, selecting a product, and quantity of selected product is defined as expected values, each in terms of a set of parameters related to customer buying decision. A likelihood function is defined for each of the set of factors. The parameters are solved using the observable data and associated likelihood function. The customer response model is time series of unit sales defined by a product combination of the expected value of customer traffic and the expected value of selecting a product and the expected value of quantity of selected product. A linear relationship is given between different products which includes a constant of proportionality that determines affinity and cannibalization relationships between the products.

CROSS REFERENCE TO RELATED PATENT APPLICATIONS

The present patent application is related to copending U.S. PatentApplication PCT/US05/19765, entitled “System and Method for ModelingCustomer Response Using Data Observable from Customer Buying Decisions”and filed concurrently herewith by Kenneth J. Ouimet et al.

FIELD OF THE INVENTION

The present invention relates in general to economic modeling and, moreparticularly, to a system and method for modeling affinity andcannibalization in customer buying decisions.

BACKGROUND OF THE INVENTION

Economic and financial modeling and planning is commonly used toestimate or predict the performance and outcome of real systems, givenspecific sets of input data of interest. An economic-based system willhave many variables and influences which determine its behavior. A modelis a mathematical expression or representation which predicts theoutcome or behavior of the system under a variety of conditions. In onesense, it is relatively easy, in the past tense, to review historicaldata, understand its past performance, and state with relative certaintythat the system's past behavior was indeed driven by the historicaldata. A much more difficult task, but one that is extremely valuable, isto generate a mathematical model of the system which predicts how thesystem will behave, or would have behaved, with different sets of dataand assumptions. While forecasting and backcasting using different setsof input data is inherently imprecise, i.e., no model can achieve 100%certainty, the field of probability and statistics has provided manytools which allow such predictions to be made with reasonable certaintyand acceptable levels of confidence.

In its basic form, the economic model can be viewed as a predicted oranticipated outcome of a mathematical expression, as driven by a givenset of input data and assumptions. The input data is processed throughthe mathematical expression representing either the expected or currentbehavior of the real system. The mathematical expression is formulatedor derived from principles of probability and statistics, often byanalyzing historical data and corresponding known outcomes, to achieve abest fit of the expected behavior of the system to other sets of data,both in terms of forecasting and backcasting. In other words, the modelshould be able to predict the outcome or response of the system to aspecific set of data being considered or proposed, within a level ofconfidence, or an acceptable level of uncertainty. As a simple test ofthe quality of the model, if historical data is processed through themodel and the outcome of the model, using the historical data, isclosely aligned with the known historical outcome, then the model isconsidered to have a high confidence level over the interval. The modelshould then do a good job of predicting outcomes of the system todifferent sets of input data.

Economic modeling has many uses and applications. One emerging area inwhich modeling has exceptional promise is in the retail salesenvironment. Grocery stores, general merchandise stores, specialtyshops, and other retail outlets face stiff competition for limitedcustomers and business. Most if not all retail stores make every effortto maximize sales, volume, revenue, and profit. Economic modeling can bea very effective tool in helping the store owners and managers achievethese goals.

Retail stores engage in many different strategies to increase sales,volume, revenue, and profit. One common approach is to offer promotionson select merchandise. The store may offer one or more of its productsat temporary sale price, discounts for multiple item purchases, orreduced service charges. One or more items may be offered with apercentage off regular price, fixed reduced price, no interestfinancing, no sales tax, or the well-known “buy two get one free” sale.The store may run advertisements, distribute flyers, and placepromotional items on highly visible displays and end-caps (end displayslocated on each isle). In general, promotional items are classified byproduct, time of promotion, store, price reduction, and type ofpromotion or offer.

The process by which retailers select and implement promotional programsvaries by season, region, company philosophy, and prior experience. Someretailers follow the seasonal trends and place on promotion those itemswhich are popular or in demand during the season. Summertime is foroutdoor activities; Thanksgiving and Christmas are for festive meals,home decorations, and gift giving; back-to-school is new clothes andclassroom supplies. Some retailers use flyers and advertisements innewspapers, television, radio, and other mass communication media forselect merchandise on promotion, without necessarily putting every itemat a reduced price. Some retailers try to call attention to certainproducts with highly visible displays. Other retailers follow thecompetition and try to out-do the other. Still other retailers utilizeloss-leaders and sell common items at cost or below cost in an effort toget customers into the store to hopefully buy other merchandise. Theretailers may also focus on which other items will sell with thepromotional items.

Promotional programs are costly and time consuming. Flyers andadvertisements are expensive to run, base margins are lost on pricereductions, precious floor-space and shelf-space are dedicated tospecific items, and significant time and energy are spent setting up andadministering the various promotions implemented by the retailer. It isimportant for the retailer to get good results, i.e. net profit gains,from the promotional investments. Yet, most if not all retailers makepromotional decisions based on canned programs, gross historicalperception, intuition, decision by committee, and other non-scientificindicators. Many promotional plans are fundamentally based on the notionthat if we did it in the past it must be good enough to do again. Inmost situations, retailers simply do not understand, or have noobjective scientific data to justify, what promotional tools are trulyproviding the best results on a time dependent per product basis.

Customers make their own buying decisions and do not necessarily followtrends. Retailers may have false understanding as to what factors havebeen primarily driving previous buying decisions and promotionalsuccesses. What has been perceived as working in the past may notachieve the same results today, possibly because the basis for belief inthe effectiveness of prior promotion programs is flawed. Other unknownor non-obvious factors may be in play driving customer buying decisionswhich undermine or reveal the weakness in previous promotions. Economic,demographic, social, political, or other unforeseen factors may havechanged and thereby altered the basis for customer buying decisions. Inspite of costly and elaborate promotions, retailers not infrequently endup with disappointing sales, lower than expected profits, unsoldinventory, and lost opportunities based on promotional guesswork. When apromotional program fails to achieve intended objectives, retailers,distributors, manufacturers, and promotional support organizations alllose confidence and business opportunity.

The retail industry as a whole operates on a low margin, high volumebusiness model. Many merchandizing elements affect customer purchasedecisions and consequently impact the volume of sales that any givenretail outlet experiences. Price and promotion are probably the two keydrivers of sales volume, with promotion playing an increasinglyimportant role as retail margins and prices measured in inflationadjusted dollars drop. The margin pressure experienced by the retailindustry is driven by many factors, most importantly the increase inlarge-box (e.g., Costco) and “every day low price” (e.g., WalMart)players in the industry.

Since margins are low, promotions almost always represent a drop inprofit generated by the promoted items, in spite of the increase insales caused by the promotional activity. Clearly, then, the motivationof retailers to participate in promotional activity is that thepromotion items become “loss-leaders”, i.e., that the promoted itemswill either drive increased traffic to the store, and/or that thepromoted items will drag other items along with the promotion purchase.This effect is termed “affinity” in the retail industry. The inverseeffect, in which sales of non-promotional items are lost due to thepurchase of promoted items in their stead is termed “cannibalization”.Characterization and prediction of the affinity and cannibalization (AC)effect are the central concerns of the method presented here.

The AC effect is widely believed in the retail industry to be asignificant driver of dollars to the enterprise bottom line.Consequently, there is significant economic value in a model which canboth characterize AC relationships and accurately predict fiscal impactof AC on promotional activity.

Past work in this area has centered on characterization of the AC effectbetween individual pairs of products. Consider two specific productsP_(i) and P_(j) in store S1. Most schemes report variations on fourbasic metrics, upon which an analyst can make an inference as to thestrength of the affinity relationship. These metrics are all simpleaggregate probabilities based on the co-occurrence of two products inmarket baskets.

$\begin{matrix}{{Confidence} = {{prob}( P_{i} \middle| P_{j} )}} \\{{{Reverse}\mspace{14mu}{Confidence}} = {{prob}( P_{j} \middle| P_{i} )}} \\{{{Affinity}\mspace{14mu}{Metric}} = \frac{{prob}( {P_{i}\bigwedge P_{j}} )}{{{prob}( P_{i} )}\;{{prob}( P_{j} )}}} \\{{Support} = {{prob}( {P_{i}\bigwedge P_{j}} )}}\end{matrix}$

These values are sufficient to infer if the actual co-occurrence issignificantly greater than the expectation of co-occurrence foruncorrelated products, and also if the relationship is asymmetric.Asymmetric relationships are typical if one of the two products is thedominant seller, e.g., electric drills drive sales of drill bits but notthe other way around, even though drill bits may be higher unit sellersthan electric drills.

An alternative approach to characterization of relationships is obtainedby examining temporal correlation of product sales, looking forcorrelated increases and decreases in sales between the hypothesized ACitems. This approach may be statistically more sound, and might yieldbetter characterizations than the previous method. However, it is alsomore computationally intense and therefore less practical given thelarge quantities of data that need to be processed to find relationshipsin retail sales data.

In order to plan effectively, e.g., choosing the optimal set of productsand promotional attributes, it is necessary to quantify the units,sales, and profit projections and baselines for the planned promotion.Thus, any approach that serves only to characterize but not predict theeffect of AC fails to fit the full requirements of a promotion planningsystem. Moreover the former method fails to capture correlations ofcannibalization because it ignores the correlation between the drivenproduct and the probability of not finding the driver product in thebasket. The latter method fails because it does not examine actualco-occurrence data, and because causes of fluctuation which areco-linear cannot be well-resolved by simple time series analysis.

A need exists for an economic model which helps retailers make effectiveand successful promotional decisions in view of customer responses.

SUMMARY OF THE INVENTION

In one embodiment, the present invention is a computer implementedmethod of modeling customer response comprising providing a linearrelationship between functions related to first and second products,wherein the linear relationship includes a constant of proportionalitybetween the functions related to the first and second products,expressing the linear relationship using a Taylor Series expansion,expressing the constant of proportionality using a first derivative termof the Taylor Series expansion, and solving for the constant ofproportionality using data observable from customer responses.

In another embodiment, the present invention is a method of providing acomputer model of customer response comprising providing a linearrelationship between functions related to first and second products,wherein the linear relationship includes a constant of proportionalitybetween the functions related to the first and second products,expressing the constant of proportionality in terms of functions relatedto the first and second products, and solving for the constant ofproportionality using data observable from customer responses, whereinthe constant of proportionality represents an affinity orcannibalization relationship between the first and second products.

In yet another embodiment, the present invention is a method of modelinga relationship between first and second products comprising providing acomputer model relating first and second products, wherein the computermodel includes a constant of proportionality between functions relatedto the first and second products, and solving for the constant ofproportionality using data observable from customer responses, whereinthe constant of proportionality represents affinity or cannibalizationrelationship between the first and second products.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram of retail business process using a promotionalmodel;

FIG. 2 is a customer buying decision tree;

FIGS. 3 a-3 b are plots of unit sales of a specific item over time;

FIG. 4 is a computer system for storing customer transactional data andexecuting the promotional model;

FIG. 5 is a graph of expected value of unit sales of product P_(i) as afunction of time;

FIG. 6 is a block diagram of creating the promotional model;

FIG. 7 is a block diagram of executing the promotional model;

FIG. 8 is a block diagram of an evaluation of a model baseline;

FIG. 9 illustrates the steps of providing a promotional model usingobservable data from customer purchase decisions;

FIG. 10 is a block diagram of retail business process using an affinityand cannibalization model; and

FIG. 11 illustrates the steps of providing affinity and cannibalizationfeatures for use with the promotional model.

DETAILED DESCRIPTION OF THE DRAWINGS

The present invention is described in one or more embodiments in thefollowing description with reference to the Figures, in which likenumerals represent the same or similar elements. While the invention isdescribed in terms of the best mode for achieving the invention'sobjectives, it will be appreciated by those skilled in the art that itis intended to cover alternatives, modifications, and equivalents as maybe included within the spirit and scope of the invention as defined bythe appended claims and their equivalents as supported by the followingdisclosure and drawings.

Economic and financial modeling and planning is an important businesstool which allows companies to conduct business planning, forecastdemand, model revenue, and optimize price and profit. Economic modelingis applicable to many businesses such as manufacturing, distribution,retail, medicine, chemicals, financial markets, investing, exchangerates, inflation rates, pricing of options, value of risk, research anddevelopment, and the like. In the face of mounting competition and highexpectations from investors, most if not all businesses must look forevery advantage they can muster in maximizing market share and profits.The ability to forecast demand, in view of pricing and promotionalalternatives, and to consider other factors which materially affectoverall revenue and profitability is vital to the success of the bottomline, and the fundamental need to not only survive but to prosper andgrow.

In particular, economic modeling is essential to businesses which facethin profit margins, such as general customer merchandise and otherretail outlets. Clearly, many businesses are keenly interested ineconomic modeling and forecasting, particularly when the model providesa high degree of accuracy or confidence. Such information is a powerfultool and highly valuable to the business.

The present discussion will consider economic modeling as applied toretail merchandising. In particular, understanding the cause and effectbehind promotional offerings is important to increasing theprofitability of the retail stores. The present invention addresseseffective modeling techniques for various promotions, in terms offorecasting and backcasting, and provides tools for a successful,scientific approach to promotional programs with a high degree ofconfidence.

In FIG. 1, retail outlet (retailer) 10 has certain product lines orservices available to customers as part of its business plan. The termsproducts and services are interchangeable in the present application.Retailer 10 may be a food store chain, general customer productretailer, drug store, discount warehouse, department store, specialtystore, service provider, etc. Retailer 10 has the ability to setpricing, order inventory, run promotions, arrange its product displays,collect and maintain historical sales data, and adjust its strategicbusiness plan. The management team of retailer 10 is held accountablefor market share, profits, and overall success and growth of thebusiness. While the present discussion will center around retailer 10,it is understood that the promotional modeling tools described hereinare applicable to other industries and businesses having similar goals,constraints, and needs. The model works for any product/service whichmay be promoted by the business. Moreover, the model can be used formany other decision processes in businesses other than retail such asdescribed above.

Retailer 10 has business or operational plan 12. Business plan 12includes many planning, analyzing, and decision-making steps andoperations. Business plan 12 gives retailer 10 the ability to evaluateperformance and trends, make strategic decisions, set pricing, orderinventory, formulate and run promotions, hire employees, expand stores,add and remove product lines, organize product shelving and displays,select signage, and the like. Business plan 12 allows retailer 10 toanalyze data, evaluate alternatives, run forecasts, and make operationaldecisions. Retailer 10 can change business plan 12 as needed. In orderto execute on business plan 12, the management team needs accurateeconomic models. In one application of the subject decision model, themethodology of the model is applied to promotional programs to helpretailer 10 make important operational decisions to increase theeffectiveness of such programs.

From business plan 12, retailer 10 provides certain observable data andassumptions, and receives back specific forecasts and predictions frompromotional model 14. The model performs a series of complexcalculations and mathematical operations to predict and forecast thebusiness functions in which retailer 10 is most interested. The outputof promotional model 14 is a report, chart, table, or other analysis 16,which represents the model's forecasts and predictions based on themodel parameters and the given set of data and assumptions. Report 16 ismade available to business plan 12 so that retailer 10 can makepromotional and operational decisions.

From time to time, retailer 10 may offer one or more of itsproducts/services on promotion. In general, a promotion relates to anyeffort or enticement made by retailer 10 to call attention to itsproduct lines, increase the attractiveness of the product, or otherwiseget the attention of its customer base, which may lead to, or increasethe likelihood of, the customer making the purchasing decision. Forexample, the product may be given temporary sale price, discount formultiple item purchases, and reduced service charges. One or more itemsmay be offered with a percentage off regular price, fixed reduced price,coupon, rebate, no interest financing, no sales tax, preferred customerdiscounts, or “buy two get one free” sale. Retailer 10 may runadvertisements and flyers in mass communication media such asnewspapers, television, and radio. Retailer 10 may place promotionalitems on highly visible displays and end-caps. Retailer 10 may sponsorpublic events, bring in celebrities, solicit testimonials, createslogans, utilize props, and piggy-back community efforts and currentevents. Promotional campaigns use any and all tasteful and appropriatemeans of calling attention to retailer 10 and its products/services. Ingeneral, promotional programs are classified by product/service, time ofpromotion, store, price reduction, and type of promotion or offer. Astore may be a single location, or a chain or logical group of stores.

The promotional programs can greatly influence the customer buyingdecision, in terms of selecting the store, selecting the product to buy,and choosing the number of items of selected products to purchase.Promotional programs are designed to draw customers into the store topurchase the promoted product. Retailer 10 will hope that customer 24also decides to buy additional merchandise, including items which areregular price. Moreover, the customer may be motivated to purchase morequantity of a given product by the promotion than they would havenormally wanted. If the promotion is “buy two get one free”, thenwhereas the customer may have purchased only one without promotion, thelogical choice becomes to buy at least two since, if they do, the thirdone is free. A similar motivation exists with multiple item discounts,such as three for a dollar or 10% discount for buying 10 or more. Thenatural customer response and behavior is to seek the best overallvalue. Customer 24 will likely place more than one item in his or herbasket if the offer is formatted in terms of a multiple item deal. Thepsychology behind creating properly formatted promotional programs is toinfluence the customer buying decision and increase the probability ofnot only selecting retailer 10's store and deciding to buy the promotedproduct, as well as other products, but also to increase the quantity ofproducts purchased. Customer response model 14 provides the forecastsand predictions which achieve precisely this desired result for retailer10. Customer 24 gets a good deal and retailer 10 receives greaterrevenue, profits, and goodwill from the customer.

For those items offered with promotion, report 16 gives retailer 10 theforecast information necessary to compare and contrast differentpromotional programs and combinations of promotional programs over time.Retailer 10 can use the forecasting features of promotional model 14 toselect promotional programs and products, set pricing, forecast demand,order inventory to meet such anticipated demand, and adjust itsstrategic planning and thinking process, all with the purpose and intentof maximizing market share, revenue, and profits. While the businessstrategy is formulated at enterprise level, the execution of thatstrategy usually occurs at the product or store level. Accordingly,promotional model 14 is one tool used by the management team of retailer10 to execute their strategy for the business.

Promotional model 14 is applied to the common problem of understandingthe mental and physical process that customers practice in making theall-important buying decision. More specifically, model 14 helpsretailer 10 predict and forecast the effect of different promotionalefforts and programs on customer demand and the resulting levels ofretail sales, revenue, volume, and profitability. Promotional model 14can also be used to backcast promotional programs, i.e. consider“what-if” scenarios of prior efforts with different driving factors.

In one embodiment, the customer buying decision or demand is based onthree factors or components: selection of store, selection of product,selection of quantity of product, which are illustrated in purchasedecision tree 20 of FIG. 2. In block 22, customer 24 decides where toshop, e.g. in store S1 or store S2. The decision to shop in store S1 orS2 depends in part on store history, growth, seasonality, visual appeal,image, current promotions, price, product selection, productavailability, prior experience, habit, quality, brand name, convenience,store layout, location, and customer preference built up over a longperiod of time. Once in store S1, customer 24 decides in block 28whether to buy an item, or not buy an item in block 30. The decision tobuy product P1 depends in part on price, promotion, seasonality, brandname, merchandising, shelf location, and the like. Other factors whichinfluence the decision to buy also include affinity, i.e. buying oneitem induces customer 24 to buy another related item, andcannibalization, i.e. decision to purchase one item disinclines customer24 to purchase another item. If customer 24 decides to buy a cordlessscrewdriver, he or she may also buy screws or an extra battery. Thepurchase of the cordless screwdriver may cause the customer to foregopurchase of a hand-operated screwdriver.

Once customer 24 decides to buy at least one product P1, then thecustomer makes a decision about how many items of product P1 topurchase, as shown in block 32. Accordingly, customer 24 places thequantity of product P1 in the shopping cart which he or she decides topurchase.

The process repeats for other products P2 and P3 in store S1. Assumingcustomer 24 is still in store S1, the customer makes a decision whetherto buy product P2 and, if in the affirmative, then he or she decides howmuch product P2 to buy; and likewise for product P3.

When customer 24 finishes shopping in store S1, the customer maypatronize store S2. The process described for purchase decision tree 20in store S1 also applies to customer buying decisions in store S2. Inblock 34, customer 24 decides whether to buy an item, or not buy an itemin block 36. Once customer 24 decides to buy at least one product P4,then the customer makes a decision about how many items of product P4 topurchase, as shown in block 38. Accordingly, customer 24 places theselected quantity of product P4 in the shopping cart.

When viewing customer buying decisions on a per product basis over time,the distribution of unit sales may look like the graph shown in FIG. 3a. Plot 40 illustrates a simplified baseline model for product P_(i) forvolume of sales over time. The baseline model represents unit sales ofproduct P_(i) with its regular price, without any promotional offer,i.e. no sale price, discounts, incentives, flyer, advertisement, orother customer enticements. Between times t₁ and t₃, the baseline modelof product P_(i) shows unit sales at an elevated level because ofseasonality, i.e. demand rises during certain time(s) of the year. Unitsales returns to normal level at time t₃. After time t₈, the baselinemodel shows unit sales of product P_(i) have fallen because of permanentincrease in regular price.

Plot 42 in FIG. 3 b illustrates a simplified promotional model ofproduct P_(i) for unit sales over time. Between times t₁ and t₂, thepromotional model of product P_(i) shows unit sales increase, over andabove the increase associated with seasonality, because of promotionalflyers or advertisement distributed at the beginning of the season.Between times t₂ and t₃, unit sales returns to the level associated withthe seasonal lift. Between times t₃ and t₄, unit sales returns to thebaseline rate. The promotional model also shows unit sales of productP_(i) increase at time t₄ because product P_(i) is placed on end-cap. Attime t₅, the product P_(i) is given temporary sale price, causing unitsales to increase even further. The sale price terminates at time t₆ andthe end-cap display is taken down at time t₇, each causing acorresponding decrease in unit sales.

The purpose of promotional model 14 is to predict or forecast (orbackcast), with reasonable certainty, the unit sales associated with orin response to one or more promotional programs, such as shown in plot42. Retailer 10 uses promotional model 14 to run report 16 from whichthe retailer makes decisions as to what promotions, and correspondingtiming of such programs, will provide the optimal and desired effectunder business plan 12. Retailer 10 may project forward in time andpredict volume of sales under one or more promotional efforts. The unitsales predictions for various promotional models will help retailer 10make well-reasoned business decisions. Alternatively, retailer 10 mayanalyze back in time to understand what may have happened if differentdecisions had been made. The backcast analysis is particularly usefulfollowing a less than successful promotion campaign to understand howthings may have been done differently. In any case, promotional model 14gives retailer 10 far more information than previously available to makegood business decisions.

In the normal course of business, retailer 10 collects a significantamount of data. Customer 24 patronizes a store, makes one or moreproduct selections, places the items in a basket, and proceeds to thecheckout counter. The contents of the basket containing one or moreproducts is a retail transaction. Most retail transactions are enteredinto a computer system.

A general purpose computer 50, as shown in FIG. 4, includes centralprocessing unit or microprocessor 52, mass storage device or hard disk54, electronic memory 56, and communication ports 58. Computer 50 runsapplication software for managing retail sales. Each product includesUniversal Product Code (UPC) or barcode label. The barcode is encodedwith a unique identification number for the product. The product isscanned over a barcode reader at the store checkout counter to read theUPC identification number. Barcode reader 59 is connected tocommunication port 58 to transfer the UPC data to computer 50. Computer50 may be part of a computer network which connects multiple barcodereaders in many stores to central computer system 64.

From the UPC data, a product database on hard disk 54 retrieves theprice for the product and any promotional initiatives. As each productfrom the customer's basket is scanned, computer 50 builds up atransaction in temporary file space on hard disk 54. Once thetransaction is complete and customer 24 has paid, the transactionbecomes a permanent record in the sales transaction log or database onhard disk 54, or as part of central computer system 64.

Another product feature which can be used by retailer 10 is radiofrequency identification tags (RFID). The RFID tag can be attached toproducts to track time dependent status such as date codes, inventory,and shelf stock. The RFID contains product information such asindividual product identification, time, and location. The RFIDinformation is transmitted to a receiving unit which interfaces with thestore's computer system. Retailer 10 can track shelf life for perishableitems and manage product rotation, ordering, inventory, and sales overtime. If a quantity of perishable product is nearing its end of shelflife, then that product is a prime candidate for promotion to move theabout-to-expire items. It is much more efficient for retailer 10 todiscount the product rather than have to destroy the merchandise.Retailer 10 will also know when inventory is low due to the promotionand needed to be restocked or order more. The location of the RFIDtagged product can be monitored to see how display location within thestore effects product sales. The time of the sale, e.g. day, month,year, is important in determining the distribution of the unit salesover time. The RFID information represents useful observable data.

The transaction log (TLOG) contains one or more line items for eachretail transaction, such as shown in Table 1. Each line item includesinformation such as store number, product number, time of transaction,transaction number, quantity, current price, profit, promotion number,and customer number. The store number identifies specific store; productnumber identifies a product; time of transaction includes date and timeof day; quantity is the number of units of the product; current price(in US dollars) can be the regular price, reduced price, or higher pricein some circumstances; profit is the difference between current priceand cost of selling the item; promotion number identifies any promotionfor the product, e.g. flyer, ad, sale price, coupon, rebate, end-cap,etc; customer number identifies the customer by type, class, region, orindividual, e.g. discount card holder, government sponsored orunder-privileged, volume purchaser, corporate entity, preferredcustomer, or special member. The TLOG data is accurate, observable, andgranular product information based on actual retail transactions withinthe store. TLOG data represents the known and observable results fromthe customer buying decision or process. The TLOG data may containthousands of transactions for retailer 10 per store S_(i) per day, ormillions of transactions per chain of stores per day.

TABLE 1 TLOG Data Cus- Store Product Time Trans Qty Price ProfitPromotion tomer S1 P1 D1 T1 1 1.50 0.20 PROMO1 C1 S1 P2 D1 T1 2 0.800.05 PROMO2 C1 S1 P3 D1 T1 3 3.00 0.40 PROMO3 C1 S1 P4 D1 T2 4 1.80 0.500 C2 S1 P5 D1 T2 1 2.25 0.60 0 C2 S1 P6 D1 T3 10 2.65 0.55 PROMO4 C3 S1P1 D2 T1 5 1.50 0.20 PROMO1 C4 S2 P7 D3 T1 1 5.00 1.10 PROMO5 C5 S2 P1D3 T2 2 1.50 0.20 PROMO1 C6 S2 P8 D3 T2 1 3.30 0.65 0 C6

A simplified example of TLOG data is shown in Table 1. The first lineitem shows that on day/time D1 (date and time), store S1 had transactionT1 in which customer C1 purchased one product P1 at 1.50. The next twoline items also refer to transaction T1 and day/time D1, in whichcustomer C1 also purchased two products P2 at 0.80 each and threeproducts P3 at price 3.00 each. In transaction T2 on day/time D1,customer C2 has four products P4 at price 1.80 each and one product P5at price 2.25. In transaction T3 on day/time D1, customer C3 has tenproducts P6 at 2.65 each, in his or her basket. In transaction T1 onday/time D2 (different day and time) in store S1, customer C4 purchasedfive products P1 at price 1.50 each. In store S2, transaction T1 withcustomer C5 on day/time D3 (different day and time) involved one productP7 at price 5.00. In store S2, transaction T2 with customer C6 onday/time D3 involved two products P1 at price 1.50 each and one productP8 at price 3.30.

The TLOG data in Table 1 further shows that product P1 in transaction T1had promotion PROMO1. For the present discussion, PROMO1 shall be afront-page featured item in a local flyer. Product P2 in transaction T1had promotion PROMO2, which is an end-cap display in store S1. ProductP3 in transaction T1 had promotion PROMO3, which is a reduced saleprice. Product P4 in transaction T2 on day/time D1 had no promotionaloffering. Likewise, product P5 in transaction T2 had no promotionaloffering. Product P6 in transaction T3 on day/time D1 had promotionPROMO4, which is a volume discount for 10 or more items. Product P7 intransaction T1 on day/time D3 had promotion PROMO5, which is a 0.50rebate. Product P8 in transaction T2 had no promotional offering. Apromotion may also be classified as a combination of promotions, e.g.flyer with sale price or end-cap with rebate.

Retailer 10 also provides additional information to the TLOG database orother logical storage area of hard disk 54 such as promotional calendarand events, store set-up, shelf location, end-cap displays, flyers, andadvertisements. For example, the information associated with a flyerdistribution, e.g. publication medium, run dates, distribution, productlocation within flyer, and advertised prices, is stored with TLOG dataon hard disk 54. The store set-up, including location of products,special displays, in-store price specials, celebrity visitations, andphysical amenities, is made available to the TLOG database.

Communication port 58 may connect by a high-speed Ethernet link tocommunication network 60. Communication network 60 may have dedicatedcommunication links between multiple computers, or an open architecturesystem such as the World Wide Web, commonly known as the Internet.Retailer 10 can access computer 50 remotely through communicationnetwork 60. Retailer 10 stores TLOG data on computer 50 or centralcomputer system 64. The TLOG data is used for a variety of functionssuch as financial reporting, inventory control, and planning. The TLOGdata is also available to use with promotional model 14, as describedhereinafter.

In one embodiment, promotional model 14 is application software orcomputer program residing on computer 50, central computer system 64, orcomputer system 66. The software is originally provided on computerreadable media, such as compact disks (CDs), or downloaded from a vendorwebsite, and installed on the desired computer. In one case, promotionalmodel 14 can be executed directly on computer 50, which may be locatedin the facilities of retailer 10. Retailer 10 interacts with computer 50through user control panel 62, which may take the form of a localcomputer terminal, to run promotional model 14 and generate report 16.Alternatively, retailer 10 uses computer system 66 to access promotionalmodel 14 remotely, e.g. through a website contained on hard disk 54.Retailer 10 can make requests of a third party who in turn runspromotional model 14 and generates report 16 on behalf of the retailer.The requests to generate report 16 may be made to the third partythrough the website or other communication medium.

Promotional model 14 operates under the premise that an expectation ofunit sales of product P_(i), over time, can be defined by statisticalrepresentations of certain component(s) or factor(s) involved incustomer purchasing decisions. The model may comprise a set of one ormore factors. The customer buying decision involves the customerresponse or behavior to a number of conditions that lead to a decisionto purchase or not purchase a given product. In one embodiment, theexpected value of units sales of product P_(i)

USpi

is defined as a multiplicative or product combination of expected valuesof customer traffic, share of product (selecting a specific product),and count (quantity) of selected product, which are factors related tothe customer buying decision. The expected value of units sales ofproduct P_(i) is given in equation (1) as:

USpi

=

TRAFFIC

SHARE

COUNT

  (1)

where:

-   -   TRAFFIC        is the expected value of total customer traffic shopping in        store S_(i)    -   SHARE        is the expected value of the probability of selecting a specific        product or group of products P_(i)    -   COUNT        is the expected value of the quantity of P_(i) purchased for        customers who bought P_(i)

The customer traffic, selecting a product, and quantity of selectedproducts form the set of factors representing components of the customerbuying decision. Each of the set of factors is defined in part using aset of causal parameters related to the customer buying decision orprocess. The set of parameters are set forth in utility functions U(t),as described hereinafter, and contained within the expected values oftraffic, share, and count. One or more of the values of the set ofparameters are functions of time. Thus, the expected values are one typeof statistical relationship which relates the set of factors to thecustomer buying decision process. The observable TLOG data, such astransaction, product, price, and promotion, is used to solve for the setof parameters contained in the utility function U(t).

For the expectation of customer traffic, i.e. the total number ofcustomers that will patronize store S_(i) in a specified time period,the expected value is defined in terms of an exponential function of theutility function parameters, given in equation (2) as:

TRAFFIC

=e^(U) ^(T) ^((t))  (2)

The traffic utility function U_(T)(t) is derived from certain attributesand parameters, as a function of time, related to customer decisionsinvolved in selecting store S_(i). With respect to equation (2), theutility function for the expectation of traffic is given in equation (3)as:U _(T)(t)=Q ₀ +αt+f _(T)(t)  (3)

where:

-   -   Q₀ is base transaction rate for store S_(i)    -   α is growth rate for store S_(i)    -   t is time period (e.g. hour, day, week)    -   f_(T)(t) is time dependent function of coefficients        f _(T)(t)=C _(TQ1)θ_(Q1)(t)+C _(TQ2)θ_(Q2)(t)+C        _(TQ3)θ_(Q3)(t)+C _(TQ4)θ_(Q4)(t)  (4)

where:

-   -   θ_(Q1)(t) is indicator function for quarter Q1    -   θ_(Q2)(t) is indicator function for quarter Q2    -   θ_(Q3)(t) is indicator function for quarter Q3    -   θ_(Q4)(t) is indicator function for quarter Q4    -   C_(TQ1) is traffic coefficient for quarter Q1    -   C_(TQ2) is traffic coefficient for quarter Q2    -   C_(TQ3) is traffic coefficient for quarter Q3    -   C_(TQ4) is traffic coefficient for quarter Q4

Coefficients C_(TQ1)-C_(TQ4) are parameters which collectively definethe driving forces behind store traffic for each time period. Theindicator function θ_(Q1)(t) has value 1 during quarter Q1 and is zerootherwise; indicator function θ_(Q2)(t) has value 1 during quarter Q2and is zero otherwise; indicator function θ_(Q3)(t) has value 1 duringquarter Q3 and is zero otherwise; indicator function θ_(Q4)(t) has value1 during quarter Q4 and is zero otherwise.

The indicator functions may represent any arbitrary set of time periods.The set represented may be periodic, e.g., day of week or month of year.The indicator functions may be recurring holidays, e.g., week beforeChristmas or Memorial day, or may represent non-contiguous non-recurringsets of time periods that have a relationship to the decision utility inthe minds of the customers.

To evaluate U_(T)(t) as per equation (3), the observable TLOG data inTable 1 is used to determine the number of transactions (baskets ofmerchandise) per store, per time period. According to the observabledata in Table 1, the number of transactions (baskets) B for day/time D1is B(D1)=3; for day/time D2 is B(D2)=1; for day/time D3 is B(D3)=2.Hence, there exists a set of observations from the TLOG data of Table 1given as observed baskets B_(obs): {B(D1), B(D2), B(D3)}, which providean objective, observable realization of the historical transactionprocess over time.

Next, a likelihood function is assigned for the expectation of customertraffic in each store S_(i). In the present discussion, the Poissondistribution is used to define the probability that a given number ofcustomers have decided to shop in store S_(i), i.e. has entered thestore with the intent to make purchase(s), as shown in equation (5):

$\begin{matrix}{{P(n)} = \frac{\langle {TRAFFIC} \rangle^{n}\mspace{11mu}{\mathbb{e}}^{- {\langle{TRAFFIC}\rangle}}}{n!}} & (5)\end{matrix}$

-   -   where: n is the number baskets B for a given time        TRAFFIC        is expectation of traffic in store S_(i), i.e. expectation of        seeing any given number of baskets B in store S_(i) on a given        day D_(i)

Other likelihood functions such as Multinomial can be used, for example,when considering market share of the products. In the present case, theprobability of the observed set of baskets P(B_(obs)), is given inequation (6) as a multiplicative or product combination of theprobability of each observation within the set of observations. Toevaluate P(B_(obs)) in equation (6), each observation of the set ofobservations {B(D1), B(D2), B(D3)} is evaluated within the Poissonfunction P(n) of equation (5), i.e. P(B(D1)), P(B(D2)), and P(B(D3)),which are given in equations (7)-(9).

$\begin{matrix}{{P( B_{obs} )} = {{P( {B( {D\; 1} )} )}*{P( {B( {D\; 2} )} )}*{P( {B( {D\; 3} )} )}}} & (6) \\{{P( {B( {D\; 1} )} )} = \frac{\langle {TRAFFIC} \rangle^{B{({D\; 1})}}\mspace{11mu}{\mathbb{e}}^{- {\langle{TRAFFIC}\rangle}}}{{B( {D\; 1} )}!}} & (7) \\{{P( {B( {D\; 2} )} )} = \frac{\langle {TRAFFIC} \rangle^{B{({D\; 2})}}\mspace{11mu}{\mathbb{e}}^{- {\langle{TRAFFIC}\rangle}}}{{B( {D\; 2} )}!}} & (8) \\{{P( {B( {D\; 3} )} )} = \frac{\langle {TRAFFIC} \rangle^{B{({D\; 3})}}\mspace{11mu}{\mathbb{e}}^{- {\langle{TRAFFIC}\rangle}}}{{B( {D\; 3} )}!}} & (9)\end{matrix}$

The likelihood function P(B_(obs)) in equation (6) is maximized usingthe Maximum Likelihood Method to evaluate the observable TLOG datawithin the Poisson probability function, and thereby solve for andcalibrate the parameters of U_(T)(t), which are unknown at the presenttime. Recall that

TRAFFIC

is a function of U_(T)(t) as defined in equation (3). P(B_(obs)) is thusa function of the parameters of U_(T)(t), i.e. Q₀, α, C_(TQ1), C_(TQ2),C_(TQ3), C_(TQ4), as per equations (3)-(4). As described above, theparameters of U_(T)(t) are contained within the factors that are relatedto the customer buying decision.

To solve for the parameters of U_(T)(t), a parameter vector V: {Q₀, α,C_(TQ1), C_(TQ2), C_(TQ3), C_(TQ4)} is defined. The goal is to resolvethe parameter vector V into a set of values which maximizes themagnitude of P(B_(obs)). The function P(B_(obs)) can be visualized as aresponse surface in n-dimensional space which has a maximum or peakvalue. The Maximum Likelihood Method is an iterative process in whichthe parameter vector V is updated each iteration until the maximum valuefor P(B_(obs)) is found.

Accordingly, an initial set of values for each parameter of the vector Vis estimated to create initial parameter vector V₀. The estimation ofinitial values for V₀ may come from historical data, or merely aneducated guess. The function P(B(D1)) is evaluated using the observedvalue B(D1) from Table 1 and the initial parameter vector V₀ to obtain afirst value for P(B(D1)). The function P(B(D2)) is evaluated using theobserved value B(D2) from Table 1 and the initial parameter vector V₀ toobtain a first value for P(B(D2)). The function P(B(D3)) is evaluatedusing the observed value B(D3) from Table 1 and the initial parametervector V₀ to obtain a first value for P(B(D3)). A first value ofP(B_(obs)) is calculated in equation (6) from the product combination ofthe first values of P(B(D1))*P(B(D2))*P(B(D3)).

Another set of values for each parameter of V is estimated, again usinghistorical data or an educated guess, to create parameter vector V₁. Thefunction P(B(D1)), given the observed value B(D1) and the parametervector V₁, is evaluated to obtain a second value for P(B(D1)). Thefunction P(B(D2)) is evaluated using the observed value B(D2) and theparameter vector V₁ to obtain a second value for P(B(D2)). The functionP(B(D3)) is evaluated using the observed value B(D3) and the parametervector V₁ to obtain a second value for P(B(D3)). A second value ofP(B_(obs)) is calculated in equation (6) from the product combination ofthe second values of P(B(D1))*P(B(D2))*P(B(D3)).

If the second value of P(B_(obs)) is greater than the first value ofP(B_(obs)), then the second value of P(B_(obs)) is closer to thesolution of finding the maximum value of P(B_(obs)). The second value ofP(B_(obs)) is a better solution than the first value, i.e. the secondestimate is moving in the correct direction, up the response surface ofP(B_(obs)) toward the optimal maximum solution. If the second value ofP(B_(obs)) is not greater than the first value of P(B_(obs)), thenanother parameter vector V₁ is estimated. The values are re-calculatedand, if necessary, re-estimated until the second value of P(B_(obs)) isgreater than the first value of P(B_(obs)).

A third set of values for each parameter of V is estimated, e.g.V₂=V₁+(V₁−V₀) or other educated guess, to create parameter vector V₂.The process of choosing iterative sets of values for the parametervector V can take a variety of approaches. There are many standard andspecialized algorithms that may be applied: steepest descent,conjugate-gradient method, Levenberg-Marquardt, and Newton-Raphson, toname a few. The function P(B(D1)) is evaluated using the observed valueB(D1) and the parameter vector V₂ to obtain a third value for P(B(D1)).The function P(B(D2)) is evaluated using the observed value B(D2) andthe parameter vector V₂ to obtain a third value for P(B(D2)). Thefunction P(B(D3)) is evaluated using the observed value B(D3) and theparameter vector V₂ to obtain a third value for P(B(D3)). A third valueof P(B_(obs)) is calculated in equation (6) from the product combinationof the third values of P(B(D1))*P(B(D2))*P(B(D3)).

If the third value of P(B_(obs)) is greater than the second value ofP(B_(obs)), then the third value of P(B_(obs)) is closer to the solutionof the maximum value of P(B_(obs)). If the third value of P(B_(obs)) isnot greater than the second value of P(B_(obs)), then another parametervector V₂ is estimated. The values are re-calculated and, if necessary,re-estimated until the third value of P(B_(obs)) is greater than thesecond value of P(B_(obs)).

The Maximum Likelihood Method repeats until the difference between thej-th value of P(B_(obs)) and the j+1-th value of P(B_(obs)) is less thanan error threshold ε, or until a stopping criteria has been reached,wherein iterative solutions of P(B_(obs)) are no longer changing by anappreciable or predetermined amount. The error threshold ε or stoppingcriteria is selected according to desired tolerance and accuracy of thesolution.

Assume that the parameter vector V has reached a solution of values forQ₀, α, C_(TQ1), C_(TQ2), C_(TQ3), C_(TQ4), using the Maximum LikelihoodMethod, which provides the maximum value for the likelihood functionusing the Poisson probability distribution P(B_(obs)). With theparameters of U_(T)(t) now known, the utility function U_(T)(t) is thuscompletely defined as per equations (3)-(4), and the expected value oftraffic is defined as per equation (2) for the purposes of the model.Each store S_(i) will have its own solution for U_(T)(t). Since U_(T)(t)is a function of time, the expected value of traffic for store S_(i) canbe modeled forward and backward in time. Equation (2) can be evaluatedfor any time t to predict the expected value of customer traffic instore S_(i).

Turning now to computing the expected value of share in equation (1). Inthis case, the share is an expectation of a probability. The expectedvalue of share is given in equation (10) as a function of exponentialtime dependent utility functions:

$\begin{matrix}{\langle {SHARE} \rangle = \frac{{\mathbb{e}}^{U_{S}{(t)}}}{1 + {\mathbb{e}}^{U_{S}{(t)}}}} & (10)\end{matrix}$

The utility function U_(S)(t) for the expected value of share is definedusing a set of parameters or attributes, as a function of time, relatedto customer decisions involved in selecting at least one product P_(i).The share utility function U_(S)(t) is defined at the product level,within store S_(i), and has specific promotions associated with thevarious products. With respect to equation (10), the utility functionfor the expectation of share is given in equations (11)-(13) as:

$\begin{matrix}{{U_{S}(t)} = {{V(t)} - {\beta( \frac{P - P_{0}}{P_{0}} )} + {f_{S}(t)}}} & (11)\end{matrix}$

where:

-   -   V(t) is time dependent base rate of sales for product P_(i)    -   β is price response for product P_(i)    -   P is price from TLOG data    -   P₀ is reference price, e.g. baseline price    -   f_(S)(t) is time dependent function of share coefficients        V(t)=V _(REG)θ_(REG)(t)+V _(PROMO1)θ_(PROMO1)(t)+V        _(PROMO2)θ_(PROMO2)(t)  (12)        f _(S)(t)=C _(SQ1)θ_(Q1)(t)+C _(SQ2)θ_(Q2)(t)+C        _(SQ3)θ_(Q3)(t)+C _(SQ4)θ_(Q4)(t)  (13)

where:

-   -   θ_(REG)(t) is indicator function for regular price    -   θ_(PROMO1)(t) is indicator function for promotion 1    -   θ_(PROMO2)(t) is indicator function for promotion 2    -   V_(REG) is the rate of sales for regular product status, i.e. no        promotion    -   V_(PROMO1) is rate of sales for first promotion    -   V_(PROMO2) is rate of sales for second promotion    -   C_(SQ1) is share coefficient for quarter Q1    -   C_(SQ2) is share coefficient for quarter Q2    -   C_(SQ3) is share coefficient for quarter Q3    -   C_(SQ4) is share coefficient for quarter Q4

Promotion coefficients β, V_(REG), V_(PROMO1), V_(PROMO2), C_(SQ1),C_(SQ2), C_(SQ3), C_(SQ4) are parameters which collectively define thedriving forces associated with rate of sales or promotional lift forproduct P_(i). The indicator function θ_(REG)(t) has value 1 during afirst time period associated with no promotion and is zero otherwise;indicator function θ_(PROMO1)(t) has value 1 during a second time periodassociated with promotion 1 and is zero otherwise; indicator functionθ_(PROMO2)(t) has value 1 during a third time period associated withpromotion 2 and is zero otherwise. For example, θ_(REG)(t) may be value1 during any day, week, or month when there is no promotion in place,i.e. regular price. V_(REG), rate of sales of product P_(i) underregular status (no promotion), is in effect when θ_(REG)(t)=1.θ_(PROMO1)(t) may be value 1 during a time period when promotion 1 is ineffect. The rate of sales of product P_(i) under promotion 1(V_(PROMO1)) factors into U_(S)(t) when θ_(PROMO1)(t)=1. θ_(PROMO2)(t)may be value 1 during a time period when promotion 2 is in effect. Therate of sales of product P_(i) under promotion 2 (V_(PROMO2)) factorsinto U_(S)(t) when θ_(PROMO2)(t)=1. One or more indicator functionsθ_(REG)(t), θ_(PROMO1)(t), and θ_(PROMO2)(t) may be enabled at any giventime.

To evaluate U_(S)(t) as per equation (11), the observable data in Table1 is used to determine the number of transactions b(t) (baskets ofmerchandise) containing at least one product P_(i), per store, per timeperiod, and the total number of transactions B(t) in store S_(i) pertime period. Equation (14) defines the relationship between b(t), B(t),and expected value of share.b(t)=B(t)

SHARE

  (14)

Define a set of observable data from Table 1, b_(OBS): {b(D1), b(D2),b(D3)}, where b(D1) is number of transactions or baskets containing atleast one product P1 for day/time D1, b(D2) is number of transactionscontaining at least one product P1 for day/time D2, and b(D3) is numberof transactions containing at least one product P1 for day/time D3. FromTable 1, the observable TLOG data, with respect to product P1, showsb(D1)=2, b(D2)=0, and b(D3)=1. The set of observable transaction dataB_(OBS): {B(D1), B(D2), D(D3)} is defined in the above discussion of thetraffic utility function. Hence, there exists from the TLOG data ofTable 1 an observable realization of the product selection process perproduct P_(i) over time.

Next, a likelihood function is assigned for the expectation of share foreach product P_(i), in each store S_(i). In the present discussion, theBinominal distribution is used to define the probability that customer24 has decided to purchase at least one product P_(i), i.e. has placedproduct P_(i) in the basket with the intent to make a purchase:

$\begin{matrix}{{P( b \middle| B )} = {\langle {SHARE} \rangle^{b}\mspace{11mu}( {1 - \langle {SHARE} \rangle} )^{B - b}\;\frac{B!}{{b!}\;{( {B - b} )!}}}} & (15)\end{matrix}$

Each of the set of observations {b(D1), b(D2), b(D3)} is evaluatedwithin the Binominal function P(b|B) of equation (15), as theprobability of b(t) given total number of transactions B(t), and setforth in equations (16)-(19).

$\begin{matrix}{{P( b_{OBS} \middle| B_{OBS} )} = {{P( {b( {D\; 1} )} \middle| {B( {D\; 1} )} )}*{P( {b( {D\; 2} )} \middle| {B( {D\; 2} )} )}*{P( {b( {D\; 3} )} \middle| {B( {D\; 3} )} )}}} & (16) \\{{P( {b( {D\; 1} )} \middle| {B( {D\; 1}\; )} )} = {\langle {SHARE} \rangle^{b{({D\; 1})}}\mspace{11mu}( {1 - \langle {SHARE} \rangle} )^{{B{({D\; 1})}} - {b{({D\; 1})}}}\;\frac{{B( {D\; 1} )}!}{{{b( {D\; 1} )}!}\;{( {{B( {D\; 1} )} - {b( {D\; 1} )}} )!}}}} & (17) \\{{P( {b( {D\; 2} )} \middle| {B( {D\; 2} )} )} = {\langle {SHARE} \rangle^{b{({D\; 2})}}\mspace{11mu}( {1 - \langle {SHARE} \rangle} )^{{B{({D\; 2})}} - {b{({D\; 2})}}}\;\frac{{B( {D\; 2} )}!}{{{b( {D\; 2} )}!}\;{( {{B( {D\; 2} )} - {b( {D\; 2} )}} )!}}}} & (18) \\{{P( {b( {D\; 3} )} \middle| {B( {D\; 3} )} )} = {\langle {SHARE} \rangle^{b{({D\; 3})}}\mspace{11mu}( {1 - \langle {SHARE} \rangle} )^{{B{({D\; 3})}} - {b{({D\; 3})}}}\;\frac{{B( {D\; 3} )}!}{{{b( {D\; 3} )}!}\;{( {{B( {D\; 3} )} - {b( {D\; 3} )}} )!}}}} & (19)\end{matrix}$

The likelihood function P(b_(OBS)|B_(OBS)) in equation (16) is maximizedusing the Maximum Likelihood Method, in a similar manner as describedabove for equations (6)-(9), to solve for the parameters U_(S)(t).Recall that <SHARE> is a function of U_(S)(t) as defined in equation(10). P(b(D1)|B(D1)) is thus a function of the parameters of U_(S)(t),i.e. β, V_(REG), V_(PROMO1), V_(PROMO2), C_(SQ1), C_(SQ2), C_(SQ3),C_(SQ4), as per equations (11)-(13). However, the parameters of U_(S)(t)are unknown at the present time.

To solve for the parameters of U_(S)(t), a parameter vector W: {β,V_(REG), V_(PROMO1), V_(PROMO2), C_(SQ1), C_(SQ2), C_(SQ3), C_(SQ4)} isdefined. The goal is to resolve the parameter vector W into a set ofvalues which maximizes the magnitude of P(b_(OBS)|B_(OBS)). The functionP(b_(OBS)|B_(OBS)) can be visualized as a response surface inn-dimensional space which has a maximum or peak value. The MaximumLikelihood Method is an iterative process in which the parameter vectorW is updated each iteration until the maximum value forP(b_(OBS)|B_(OBS)) is found.

Accordingly, an initial set of values for each parameter of the vector Wis estimated to create initial parameter vector W₀. The estimation ofinitial values for W₀ may come from historical data, or merely aneducated guess. The function P(b(D1)|B(D1)) is evaluated using theobserved values b(D1) and B(D1) from Table 1 and the initial parametervector W₀ to obtain a first value for P(b(D1)|B(D1)). The functionP(b(D2)|B(D2)) is evaluated using the observed values b(D2) and B(D2)from Table 1 and the initial parameter vector W₀ to obtain a first valuefor P(b(D2)|B(D2)). The function P(b(D3)|B(D3)) is evaluated using theobserved values b(D3) and B(D3) from Table 1 and the initial parametervector W₀ to obtain a first value for P(b(D3)|B(D3)). A first value ofP(b_(OBS)|B_(OBS)) is calculated in equation (16) from the productcombination of the first values ofP(b(D1)|B(D1))*P(b(D2)|B(D2))*P(b(D3)|B(D3)).

Another set of values for each parameter of W is estimated, again usinghistorical data or an educated guess, to create parameter vector W₁. Thefunction P(b(D1)|B(D1)) is evaluated using the observed values b(D1) andB(D1) and the parameter vector W₁ to obtain a second value forP(b(D1)|B(D1)). The function P(b(D2)|B(D2)) is evaluated using theobserved values b(D2) and B(D2) and the parameter vector W₁ to obtain asecond value for P(b(D2)|B(D2)). The function P(b(D3)|B(D3)) isevaluated using the observed values b(D3) and B(D3) and the parametervector W₁ to obtain a second value for P(b(D3)|B(D3)). A second value ofP(b_(OBS)|B_(OBS)) is calculated in equation (16) from the productcombination of the second values ofP(b(D1)|B(D1))*P(b(D2)|B(D2))*P(b(D3)|B(D3)).

If the second value of P(b_(OBS)|B_(OBS)) is greater than the firstvalue of P(b_(OBS)|B_(OBS)), then the second value of P(b_(OBS)|B_(OBS))is closer to the solution of finding the maximum value ofP(b_(OBS)|B_(OBS)). The second value of P(b_(OBS)|B_(OBS)) is a bettersolution than the first value, i.e. the second estimate is moving in thecorrection direction, up the response surface of P(b_(OBS)|B_(OBS))toward the optimal maximum solution. If the second value ofP(b_(OBS)|B_(OBS)) is not greater than the first value ofP(b_(OBS)|B_(OBS)), then another parameter vector W₁ is estimated. Thevalues are re-calculated and, if necessary, re-estimated until thesecond value of P(b_(OBS)|B_(OBS)) is greater than the first value ofP(b_(OBS)|B_(OBS)).

A third set of values for each parameter of W is estimated, e.g.W₂=W₁+(W₁−W₀) or other educated guess, to create parameter vector W₂.Again, the process of choosing iterative sets of values for theparameter vector W can take a variety of approaches: steepest assent,gradient method, Levenberg-Marquadt, Newton-Raphson. The functionP(b(D1)|B(D1)) is evaluated using the observed values b(D1) and B(D1)and the parameter vector W₂ to obtain a third value for P(b(D1)|B(D1)).The function P(b(D2)|B(D2)) is evaluated using the observed values b(D2)and B(D2) and the parameter vector W₂ to obtain a third value forP(b(D2)|B(D2)). The function P(b(D3)|B(D3)) is evaluated using theobserved values b(D3) and B(D3) and the parameter vector W₂ to obtain athird value for P(b(D3)|B(D3)). A third value of P(b_(OBS)|B_(OBS)) iscalculated in equation (16) from the product combination of the thirdvalues of P(b(D1)|B(D1)))*P(b(D2)|B(D2))*P(b(D3)|B(D3)).

If the third value of P(b_(OBS)|B_(OBS)) is greater than the secondvalue of P(b_(OBS)|B_(OBS)), then the third value of P(b_(OBS)|B_(OBS))is closer to the solution of the maximum value of P(b_(OBS)|B_(OBS)). Ifthe third value of P(b_(OBS)|B_(OBS)) is not greater than the secondvalue of P(b_(OBS)|B_(OBS)), then another parameter vector V₂ isestimated. The values are re-calculated and, if necessary, re-estimateduntil the third value of P(b_(OBS)|B_(OBS)) is greater than the secondvalue of P(b_(OBS)|B_(OBS)).

The Maximum Likelihood Method repeats until the difference between thej-th value of P(b_(OBS)|B_(OBS)) and the j+1-th value ofP(b_(OBS)|B_(OBS)) is less than an error threshold ε, or until astopping criteria has been reached, wherein iterative solutions ofP(b_(OBS)|B_(OBS)) are no longer changing by an appreciable orpredetermined amount. The error threshold ε or stopping criteria isselected according to desired tolerance and accuracy of the solution.

Assume that the parameter vector W has reached a solution of values forβ, V_(REG), V_(PROMO1), V_(PROMO2), C_(SQ1), C_(SQ2), C_(SQ3), C_(SQ4),using the Maximum Likelihood Method, which provides the maximum valuefor the likelihood function P(b_(OBS)|B_(OBS)). With the parameters ofU_(S)(t) now known, the utility function U_(S)(t) is thus defined as perequation (11), and the expected value of share is defined as perequation (10). Each product P_(i) will have its own solution forU_(S)(t). Since U_(S)(t) is a function time, the expected value of sharefor product P_(i) can be modeled forward and backward in time. Equation(10) can be evaluated for any time t to predict the expected value ofshare for each product P_(i).

Turning to computation of the expected value of count in equation (1).In one embodiment, the expected value of count is an average of thequantity of items of product P_(i) in those baskets containing at leastone product P_(i), given in equation (20) as:

$\begin{matrix}{\langle {COUNT} \rangle = \frac{\sum\limits_{t = 1}^{N}{C(t)}}{N}} & (20)\end{matrix}$

Alternatively, the expected value of count is defined in terms of aprobability distribution. In equation (21), P_(a) is the probability ofaccepting one more item of product P_(i), given that customer 24 hasalready selected at least one product P_(i). In equation (22), theexpected value of count is given in terms of P_(a).

$\begin{matrix}{P_{a} = \frac{{\mathbb{e}}^{U_{C}{(t)}}}{1 + {\mathbb{e}}^{U_{C}{(t)}}}} & (21) \\{\langle {COUNT} \rangle = \frac{1}{1 - P_{a}}} & (22)\end{matrix}$

The utility function U_(C)(t) for the expected value of count is definedusing attributes and parameters, as a function of time, related tocustomer decisions involved in selecting a specific quantity of productP_(i), given that customer 24 has chosen to purchase at least oneproduct P_(i). The count utility function U_(C)(t) is defined at theproduct level, within store S_(i), and has specific promotionsassociated with the various products. With respect to equation (21), theutility function for the expectation of count is given in equations(23)-(25) as:

$\begin{matrix}{{U_{C}(t)} = {{V(t)} - {\beta\;( \frac{P - P_{0}}{P_{0}} )} + {f_{C}(t)}}} & (23)\end{matrix}$

-   -   where: f_(C)(t) is time dependent function of share coefficients        V(t)=V _(REG)θ_(REG)(t)+V _(PROMO1)θ_(PROMO1)(t)+V        _(PROMO2)θ_(PROMO2)(t)  (24)        f _(C)(t)=C _(CQ1)θ_(Q1)(t)+C _(CQ2)θ_(Q2)(t)+C        _(CQ3)θ_(Q3)(t)+C _(CQ4)θ_(Q4)(t)  (25)

where:

-   -   C_(CQ1) is share coefficient for quarter Q1    -   C_(CQ2) is share coefficient for quarter Q2    -   C_(CQ3) is share coefficient for quarter Q3    -   C_(CQ4) is share coefficient for quarter Q4

Promotion coefficients β, V_(REG), V_(PROMO1), V_(PROMO2), C_(CQ1),C_(CQ2), C_(CQ3), C_(CQ4) are parameters which collectively define thedriving forces associated with rate of sales or promotional lift forproduct P_(i). To evaluate U_(C)(t) as per equation (23), the observabledata in Table 1 is used to determine the expected value of the count ofproduct P_(i) in a transaction containing at least one product P_(i) perstore, per time period.

Define a set of observable data from Table 1 for product P_(i), C_(OBS):{C(D1,T1), C(D1,T2), C(D1,T3), C(D2,T1), C(D3,T1), C(D3,T2)}, whereC(D_(i),T1) is the count of items of product P_(i) in the basket fortransaction T1 for time D_(i), C(D_(i),T2) is the count of items ofproduct P_(i) in the basket for transaction T2 for time D_(i), andC(D_(i),T3) is the count of items of product P_(i) in the basket fortransaction T3 for time D_(i). From Table 1, the observable data, withrespect to product P1, shows C(D1,T1)=1, C(D1,T2)=4, C(D2,T1)=5, andC(D3,T2)=2. P(c) for other products P_(i) and sets of observable dataare calculated in a similar manner. Hence, there exists from the TLOGdata of Table 1 an observable realization of the count process forproduct P_(i) over time.

Next, a likelihood function is assigned for the expectation of count foreach product P_(i), in each store S_(i). The probability of having citems of product P_(i) in the basket, given that there is at least oneitem of product P_(i) in the basket, is given in equation (26):P(c)=P _(a) ^(c−1)(1−P _(a))  (26)

The probability of seeing each of the set of observables C_(OBS) isevaluated within the P(c) in equations (27)-(31) as:P(C _(OBS))=P(C(D1,T1))*P(C(D1,T2))*P(C(D2,T1))*P(C(D3,T2))  (27)P(C(D1,T1))=P _(a) ^(C(D1,T1)−1)(1−P _(a))  (28)P(C(D1,T2))=P _(a) ^(C(D1,T2)−1)(1−P _(a))  (29)P(C(D2,T1))=P _(a) ^(C(D2,T1)−1)(1−P _(a))  (30)P(C(D3,T2))=P _(a) ^(C(D3,T2)−1)(1−P _(a))  (31)

The likelihood function P(C_(OBS)) in equation (27) is maximized usingthe Maximum Likelihood Method, in a similar manner as described abovefor P(B_(OBS)) in equation (6) and P(b_(OBS)|B_(OBS)) in equation (16),to solve for the parameters of U_(C)(t). P(C_(obs)) is a function of theparameters of U_(C)(t), i.e. β, V_(REG), V_(PROMO1), V_(PROMO2),C_(CQ1), C_(CQ2), C_(CQ3), C_(CQ4), as per equation (21). The MaximumLikelihood resolves the parameter vector X: {β, V_(REG), V_(PROMO1),V_(PROMO2), C_(CQ1), C_(CQ2), C_(CQ3), C_(CQ4)} associated with thecount utility function U_(C)(t) using a similar iterative process, untilthe difference between the j-th value of P(C_(obs)) and the j+1-th valueof P(C_(obs)) is less than an error threshold ε, or a stopping criteriahas been reached, wherein iterative solutions of P(C_(obs)) are nolonger changing by an appreciable amount. The error threshold ε orstopping criteria is selected according to desired tolerance andaccuracy of the solution.

Assume that the parameter vector X has reached a solution of values forβ, V_(REG), V_(PROMO1), V_(PROMO2), C_(CQ1), C_(CQ2), C_(CQ3), C_(CQ4),using the Maximum Likelihood Method, which provides the maximum valuefor the likelihood function P(C_(obs)). With the parameters for U_(C)(t)now known, the utility function U_(C)(t) is thus defined as per equation(23), and the expected value of count is defined as per equations (21)and (22). Each product P_(i) will have its own solution for U_(C)(t).Since U_(C)(t) is a function time, the expected value of count forproduct P_(i) can be modeled forward and backward in time. Equation (22)can be evaluated for any time t to predict the expected value of countfor product P_(i).

With the expected values of customer traffic, share, and countreevaluated in terms of a plurality of parameters, the promotional model14 is defined as the expected value of unit sales for product P_(i) asper equation (1). The promotional model 14 gives retailer 10 the abilityto forecast or predict economic models, given proposed sets of inputdata or what-ifs, which have significant impact on its business.

A set of graphs for unit sales of one product P_(i)

USp_(i)

as a function of time is shown in FIG. 5. Plot 72 represents actual unitsales of product P_(i) sold in store S_(i) over the time period. Plot 74represents forecast of product P_(i) under promotion using promotionalmodel 14. Plot 76 represents baseline (no promotion) of product P_(i).Plot 78 represents a moving average of units sales of product P_(i) fromplot 72. FIG. 5 illustrates the time series of the expected value ofunit sales for product P_(i)

USp_(i)

has been defined in equation (1) in terms of a product combination ofexpected values of factors (traffic, share, count) which influence thecustomer buying decision. Report 16 may include FIG. 5 as part of itsforecast information.

The time series of

USp_(i)

illustrates a number of promotional lifts, volume declines, andgenerally a significant amount of variation in unit sales of productP_(i). Promotion model 14 also provides additional information toretailer 10, in the form of a table or chart, including (1) baseline ofproduct P_(i), such as shown in plot 40, (2) observables B_(OBS),b_(OBS)|B_(OBS), and C_(OBS) determined from Table 1, and (3) values ofthe drivers of

USp_(i)

, in terms of the parameters of U_(T)(t), U_(S)(t), and U_(C)(t). Theadditional information is used to resolve and understand the promotionalvariation per unit time period seen in FIG. 5. For example, retailer 10can analyze the difference between V_(REG) and V_(PROMO) at specificpoints in time to help explain movement in the model.

In another embodiment, combinations of one or more of the expectedvalues of customer traffic, share, and count can be modeled individuallyor collectively. For example, model 14 may consider the expected valueof customer traffic alone, or the expected value of share alone, theexpected value of count alone, or the combination of expected values ofcustomer traffic and share. The orientation of the model will bedetermined in part by the forecast request made by retailer 10.

The above discussion has considered the TLOG data from Table 1 as oneembodiment of the data observable from customer responses. Theobservable data can originate from other sources and with differentcharacteristics, format, and content. In general, the observable datacan be categorized as customer response data, causal data, and otherdata which influences customer buying decisions. Customer response dataincludes the TLOG data from the checkout counter of customer purchasesin the basket, as discussed above in the solution of the expected valuesof traffic, share, and count. Other sources of customer response datainclude direct and indirect observations or detection of the customerinterest in one or more products. The detection of customer 24 interestin product P_(i) may be made by observing the customer handle theproduct. If a particular display or shelf storage of product P_(i) isregularly found in disarray, even after housekeeping, then it isreasonable to conclude that the customers have been showing sufficientinterest to handle the product and read the label, ingredients, andbenefits. The store may even use motion sensors to detect where and forhow long customers are spending their time. Individual products,especially high price items, may be given sensors to detect when eachproduct has been handled by the customer for consideration of purchase.The store camera system may detect congregations of customers aroundcertain products or displays over time. The camera system can detect eyemovement of the customers to see if they are focusing on certainproducts. Even through the customer did not select the product forpurchase, the interest in the product and probability of futurepurchases may be high.

Another source of customer response data is customer surveys and bodycounts. The store may entice customers with coupons, gifts, and prizesto participate in surveys which ask what product(s) they have come toshop for, interest in specific products, and their opinions on the storeand its products. The store may electronically count customers enteringand leaving the store over time to get a temporal distribution of storetraffic. The store camera system or sensors may detect how manycustomers are in each isle or part of the store over time.

Causal data is another form of observable data. Causal data relates tovarious attributes that cause or induce the customer buying decision.Product attributes include brand, size, color, and form factor. Locationattributes involve store, website, and catalog. Customer attributesinclude loyalty cards and demographics. Promotional attributes includeadvertisement, price, and incentives. Competitor data considers relativemarket share, competing price and promotions, store density pergeographic area, and product assortment. Calendar data may involvemerchandising; e.g. store layout, allocation of space, and displaytiming; inventory; promotion and price calendar; and demand planning,e.g. holidays, customer cycles, special events, weather, store densityper geographic area. Credit card history and loyalty cards providesignificant information related to store and product selections. Priorsis another type of data which provides information about the expecteddistribution of parameter values within the model, which may be based oninformation from prior purchases of similar or different products, meanand standard deviation data from prior transactions, and general trendsin the relevant marketplace.

The different types of observable data, customer response data, causaldata, and other data related to the customer buying decision, providesignificant insight into determining the customer's interests,preferences, likes, dislikes, quirks, and responses to store promotion,layout, product selection, presentation, and arrangement. The process ofcollecting and analyzing all types of observable data is important toservicing the customer's needs and increasing store profitability.

The observable data can readily be applied to solving parameters whichdefine promotional model 14. As previously discussed, the modelcomprises one or more factors which relate to the customer buyingdecision. The factors contain one or more parameters which define thebehavior of the model. All of the types of data observable from thecustomer buying decision, as discussed above, can be used to solve theparameters which in turn completes the model.

FIG. 6 illustrates a summary of creating promotional model 14. Block 80represents observable data, e.g. customer response data, causal data,priors. Block 82 performs a segmentation process on the observable datato prepare the data for presentation to the modeling program. Thesegmentation process filters and separates the observable data bylocation, product, promotion, time, customers, etc., based on a set oflogical rules applied to one or more of the data elements included ineach transaction. These rules may include restricting the store,product, customer, price, or promotion values to a logical subset of thepossible values. The filters may also restrict the customers to thosebelonging to a specific customer profile. The net result is that themodels generated from the segmented data will describe the customerbehavior displayed by that particular market segment. Such informationmay be of great value to retailers seeking to enhance performancethrough marketing to customer segments or identifying businessstrategies based on knowledge of customer information.

Block 84 performs an aggregation of the segmented data to prepare thedata for modeling. The aggregation phase may transform input data at therawest granularity to a higher granularity in accordance with thesegmentation groupings outlined above. The transformation may includesimple summations over the groups, or a more complex functional orlogical form. Throughout the discussion, “store” is understood to referto one of the following: a single sales outlet, location or channel, aspecified set of outlets, locations or channels, or a logical groupingof outlets, locations or channels. Similarly, “product” is understood torefer to a single item or service, a specific set of items and/orservices, a logical grouping of items or services. The “product” maycorrespond to a single scan item in a Point of Sale system, or it maycorrespond to multiple scan items. Customers and promotions may besimilarly grouped and labeled for the purposes of data processing and/ormodeling. Time periods are arbitrary but presumed uniform, e.g., hour,day, week, etc. The aggregation process allows observable data to begrouped within stable and relatively constant causal indicators.

Block 86 calibrates the model by solving for the unknown parameters.Block 88 stores the parameters on hard disk 54 or other accessibledatabase for later use by customer response model 14.

In FIG. 7, the proposed model data (proposed prices, promotions, timeperiod, etc.) and solved model parameters from blocks 90 and 92 areexecuted by the model. Customer response model 14 runs the forecast inblock 94 using the proposed model data and generates report 16 in block96.

In FIG. 8, a promotional evaluation builds a model baseline in block 100using the solved parameters and historical data. The model baselinerepresents expected sales without promotion. Block 102 compares themodel baseline to actuals to determine lift in unit sales attributed tothe promotion. Block 104 generates report 16.

The process of forecasting a promotional model using observable datafrom customer purchase decisions is shown in FIG. 9. Step 120 providesdata observable from customer responses. The observable data includestransaction, product, price, and promotion. Step 122 provides anexpected value of customer traffic within a store in terms of a firstset of parameters. The expected value of customer traffic is anexponential function of the first set of parameters. A likelihoodfunction using the Poisson distribution is also defined for the customertraffic. Step 124 solves the first set of parameters through aniterative estimation process using the likelihood function and theobservable data. Step 126 provides an expected value of selecting aproduct in terms of a second set of parameters. The expected value ofselecting a product is comprised of exponential functions of the secondset of parameters. A likelihood function using the Binominaldistribution is also defined for selecting a product. Step 128 solvesthe second set of parameters using the likelihood function and theobservable data. Step 130 provides an expected value of quantity ofselected product in terms of a third set of parameters. A likelihoodfunction is also defined for the quantity of selected product. Step 132solves for the third set of parameters using the likelihood function andthe observable data. Step 134 provides a promotional model as a productcombination of the expected value of customer traffic and the expectedvalue of selecting a product and the expected value of quantity ofselected product. The promotional model is used to generate report 16for retailer 10.

The analysis of report 16, as generated by model 14, helps explain theeffect of promotional programs on unit sales, revenue, andprofitability. Retailer 10 can use promotional model 14 to forecast unitsales per product over time, under a variety of promotional programs,and combinations of promotional programs. Understanding the cause andeffect behind promotional offerings is important to increasing theprofitability of the retail stores. The promotional model addresseseffective analysis techniques for various promotions, in terms offorecasting and backcasting, and provides tools for a successful,scientific approach to promotional programs with a high degree ofconfidence. Although unit sales in the retail environment has beendiscussed in detail as the economic model, promotional model 14 isapplicable to many other economic, scientific, and commercial decisionprocesses. The model can be used for business planning, assessing avariety of what-if scenarios, business optimization, and evaluation ofhistorical events to compare actuals to forecast and actuals to baselinedata.

Another economic model which is useful in the retail business isdescribed in FIG. 10. Components of FIG. 10 having a similar functionare assigned the same reference number used in FIG. 1. Retailer 10 usesbusiness plan 12 to collect and analyze transaction data, evaluatealternatives, run forecasts, and make operational decisions. Frombusiness plan 12, retailer 10 provides certain observable data andassumptions, and receives back specific forecasts and predictions fromaffinity and cannibalization model 140. The model performs a series ofcomplex calculations and mathematical operations to predict and forecastthe effect on product P_(i) attributed to product P_(j) which is onpromotion. Product P_(i) may or may not be on promotion. The affinityand cannibalization model 140 may also interact with promotional model14 to exchange, format, and evaluation data. The output of affinity andcannibalization model 140 is a report, chart, table, or other analysis142, which represents the model's forecasts and predictions based on thegiven set of data and assumptions. Report 142 is made available tobusiness plan 12 so that retailer 10 can make promotional andoperational decisions.

The method presented herein attempts to accurately predict the fiscalimpact of affinity and cannibalization (AC) with a practical andscalable solution applicable to the complexities and data volumes foundin the retail industry. The features of a practical system forestimating the fiscal impact of AC are: (1) model for cause and effectof changes in sales of individual products, (2) model relatingcorrelated changes in sales of pairs or groups of products, (3)observable data including transaction line-item detail to calibratemodels, and (4) flexible processing system to process data, generate,and evaluate the models.

In the overall process, a first function picks up the TLOG data andinputs it into a segmentation function described above. The segmentationfunction essentially filters the data according to logical rules. Aftersegmentation, the data may be aggregated to any logical groupings in theprimary dimensions: store, product, time, customer, and promotion.Finally, a process extracts the relevant co-occurrence data within thesegment and logical groupings discussed above. Co-occurrence datadescribes the number and frequency of transaction events containing bothproducts and each product individually. The data is input into themodeling process, which both characterizes the discovered relationshipsand also calibrates the model. The modeling process may be run multipletimes for various market segments and aggregation levels. The output ofeach run is a set of parameter values which characterizes ACrelationships, and is input to a forecasting process which generatesestimates (forecasts and/or backcasts) of the fiscal impact of the ACeffect.

Two differences between the present method and the other methodsdiscussed in the background is the ability to generate estimates offiscal impact, and also to characterize and estimate cannibalizationrelationships. In addition, the method captures the drivers of cause andeffect at both the product level and the relationship level. Thesedrivers include but are not limited to seasonality, price, promotion,product availability (inventory, product introduction/discontinuation),product visibility (merchandizing), and competitor strategy (priceimage). These drivers may be captured through the modeling process,which can be based on powerful Maximum Likelihood methods for nonlinearparameter estimation, and provides a rigorous framework for estimatingthe impact of 2-body product interactions. The framework supports theincorporation of non-observable data such as prior information ofparameter probability distributions. Finally, the method is flexibleenough to accept as input and leverage any methods for marketsegmentation of data and aggregation for strategic reporting,evaluation, and decision support.

It is possible for sales of product P_(j) to induce sales of anotherproduct P_(i), or for sales of product P_(j) to hinder sales of anotherproduct P_(i). Product P_(j) is considered the driver product andproduct P_(i) is the product being driven. When a sale of product P_(j)causes or induces customer 24 to also buy product P_(i), therelationship is referred to as affinity. When a sale of product P_(j)impedes or causes customer 24 to forgo purchase of product P_(i), therelationship is referred to as cannibalization. Accordingly, theaffinity or cannibalization relationship is a linkage between productsP_(i) and P_(j). In general, the relationship between product P_(i) andP_(j) will be either affinity or cannibalization, or there may be nolinkage. The processes of affinity and cannibalization are important tounderstand when modeling unit sales of products, and for understandingthe impact which a promotion assigned to one product may have on unitsales of another product.

Consider the example of the cordless drill. Customer 24 may make thedecision to purchase the cordless drill to use in various home projects,or as part of his or her business. Cordless drills are adapted toreceive drill bits, screwdriver heads, and other attachments. Cordlessdrills have the advantage and convenience of drawing power from abattery source, which means there is no electrical cord to plug into anelectrical outlet. The purpose of the cordless drill is to drill holesand, with the proper attachments, to screw fasteners into buildingmaterials. When customer 24 makes the decision to purchase the cordlessdrill, he or she will consider the variety of options also available forpurchase. Customer 24 may need drill bits, screwdriver head, screwdriverbits, socket set, screws, and extra batteries. Cordless drill can alsooperate attachments such as alignment tool, grinder, wire brush, andbuffer, which customer 24 may decide to buy. Retailer 10 may also offera tool chest to contain all the components, home improvement classes,and extended warranty service programs. Once customer 24 decides to buythe cordless drill, then the stage is set for projects which make use ofthe tool, e.g. building shelves, installing ceiling fans, finishing apatio, replacing door hinges, just to name a few. Accordingly, thepurchase of the cordless drill creates an affinity for otherproducts/services.

On the other hand, the cordless drill also has the opposite effect onother purchases. If customer 24 decides to purchase the cordless drill,he or she will likely forgo purchase of a drill having an electricalcord. Moreover, customer 24 may decide to not purchase a hand-heldscrewdriver since the cordless drill will likely do the same job in lesstime and with greater ease. The purchase of the cordless drill maycreate a cannibalization effect for other products/services.

The effects of affinity and cannibalization are even greater whenproduct P_(j) is placed on promotion. A promotion of product P_(j)usually increases its unit sales. When product P_(j) is placed onpromotion and experiences an increase in unit sales as projected, thenthe lift in sales of product P_(j) due the promotion causes acorresponding increase in sales of product P_(i) by the affinitylinkage. The product P_(i) having the affinity relationship with productP_(j) should be modeled to understand the relationship and its effect onstore operations. Retailer 10 should factor the increase in sales ofproduct P_(i), caused by the promotion of product P_(j), into businessplan 12 to plan promotions, accurately forecast revenue, and ensure thatsufficient product P_(i) is on-hand to meet customer demand.

Likewise, when product P_(j) is placed on promotion and experiences anincrease in unit sales as projected, then the increase in sales ofproduct P_(j) from the promotion may also cause a corresponding decreasein sales of product P_(i) by any cannibalization linkage. The productP_(i) having the cannibalization relationship with product P_(j) shouldbe modeled to understand the relationship and its effect on storeoperations. Retailer 10 should factor the decrease in sales of productP_(i), caused by the promotion of product P_(j), into business plan 12to plan promotions, accurately forecast revenue, and ensure that productP_(i) does not end up over-stocked. Thus, the promotion of product P_(j)has multi-faceted impact on products P_(i) and P_(j).

From the discussion of the share model, b_(i)(t) is the number oftransactions (baskets of merchandise) containing at least one productP_(i) per store, per time period. The basket share model b_(j)(t) is thenumber of transactions containing at least one product P_(j) per store,per time period. The affinity and cannibalization model usesco-occurrence of products P_(i) and P_(j) in the same basket, asobserved from the TLOG data, to estimate relationship between the pairof products. If products P_(i) and P_(j) are uncorrelated, then they maybe found together in the same transaction with a certain baseprobability. If the observed occurrence of products P_(i) and P_(j)being found together in the same transaction is greater than theuncorrelated base probability by statistically significant amount, thenan affinity relationship exists between the products. The purchase ofproduct P_(j) influences the buying decision for customer 24 to alsopurchase product P_(i). If the observed occurrence of products P_(i) andP_(j) being found together in the same transaction is less than theuncorrelated base probability by statistically significant amount, thena cannibalization relationship exists between the products. The purchaseof product P_(j) influences the buying decision for customer 24 to forgopurchase of product P_(i).

An observed increased (decreased) probability of co-occurrence is notsufficient to quantify the affinity (cannibalization) relationshipbetween the products. Typically the relationship will not be symmetricand one product will be the dominant, or driver, product.

The relationship between changes of basket share models b_(i)(t) andb_(j)(t) is shown in equation (32). We define the change in the basketshare models, Δb_(i)(t) and Δb_(j)(t), of products P_(i) and P_(j) dueto a change in the price or promotion state of the driver product Pj asthe model share values before the change minus the model share valuesafter the change. The change in basket share models Δb_(i)(t) andΔb_(j)(t) are functions of products P_(i) and P_(j).Δ(b _(i)(t))=α_(ij)Δ(b _(j)(t))  (32)

The linear relationship in equation (32) shows that the change inbaskets of b_(j)(t) is proportional to change in baskets of b_(i)(t) bythe constant of proportionality (α_(ij)) between products P_(i) andP_(j). In other words, the linear relationship between b_(i)(t) andb_(j)(t) relates baskets of the first product to baskets of the secondproduct by the constant of proportionality. Consider the example wheretransaction counts of product P_(j) increases by 10 per store per daydue to a promotional lift. If the constant of proportionality α_(ij)between products P_(i) and P_(j) is 0.1, then retailer 10 can expect tohave 1 more transaction containing product P_(i) per store per day, dueto the promotion of product P_(j).

The linear relationship between changes in basket share models can bedefined in terms of the following conditional probabilityΓ(P_(i)|P_(j)), see equations (33)-(34).b _(i)=Γ(P _(i) |P _(j))b _(j)+Γ(P _(i)| P _(j) ) b _(j)   (33)b _(i)=Γ(P _(i) |P _(j))b _(j)+Γ(P _(i)| P _(j) )(1−b _(j))  (34)

The conditional probability Γ(P_(i)|P_(j)) is total baskets containingproducts P_(i) and P_(j) over total baskets containing product P_(j). Itis interpreted as the probability of purchasing P_(i) given that P_(j)is already selected for purchase. The conditional probability Γ(P_(i)|P_(j) ) is total baskets containing product P_(i) and no product P_(j)over total baskets containing no product P_(j). The basket share modelb_(j) is the number of transactions (baskets of merchandise) containingno product P_(j) per store, per time period.

As an approximation toward the solution of constant of proportionalityα_(ij), assume that the conditional probabilities Γ(P_(i)|P_(j)) andΓ(P_(i)| P_(j) ) are independent of basket share model b_(j). Equation(32) is then derived from a Taylor series expansion:

$\begin{matrix}{{f(x)} = {{f( x_{0} )} + {{f^{\prime}(x)}( {x - x_{0}} )} + \frac{{f^{\prime\prime}(x)}( {x - x_{0}} )^{2}}{2!} + \ldots\mspace{11mu} + \frac{{f^{(n)}(x)}( {x - x_{0}} )^{n}}{n!} + {O( {x - x_{0}} )}^{n + 1}}} & (35)\end{matrix}$of the functional dependence of b_(i) on b_(j). The Taylor series isevaluated at the non-promotional state of b_(i). Each term in theexpansion represents a deviation from this baseline state. The earlyterm(s) in the Taylor series can be used to estimate and define theconstant of proportionality. For example, the first derivative term ofthe Taylor series represents the slope of the function at x₀. The firstderivative can be used to estimate α_(ij), as shown in equations(36)-(38). Higher order terms can be used as well.

$\begin{matrix}{\alpha_{ij} = \frac{\partial b_{i}}{\partial b_{j}}} & (36) \\{\alpha_{ij} = \frac{\partial( {{{\Gamma( P_{i} \middle| P_{j} )}b_{j}} + {{\Gamma( P_{i} \middle| \overset{\_}{P_{j}} )}( {1 - b_{j}} )}} )}{\partial b_{j}}} & (37) \\{\alpha_{ij} = {{\Gamma( P_{i} \middle| P_{j} )} - {\Gamma( P_{i} \middle| \overset{\_}{P_{j}} )}}} & (38)\end{matrix}$

The constant of proportionality α_(ij) can now be determined from theTLOG data. Consider the following set of TLOG data in Table 2:

TABLE 2 TLOG Data Cus- Store Product Time Trans Qty Price ProfitPromotion tomer S1 P1 D1 T1 1 1.50 0.20 PROMO1 C1 S1 P1 D1 T2 1 1.500.20 PROMO1 C2 S1 P2 D1 T2 1 3.00 0.40 0 C2 S1 P2 D1 T3 1 3.00 0.40 0 C3S1 P3 D1 T4 1 2.25 0.60 0 C4 S1 P4 D1 T5 1 2.65 0.55 0 C5 S1 P5 D1 T6 11.55 0.20 0 C6 S1 P6 D1 T7 1 5.00 1.10 0 C7 S1 P7 D1 T8 1 1.90 0.50 0 C8S1 P8 D1 T9 1 3.30 0.65 0 C9 S1 P9 D1 T10 1 2.30 0.55 0 C10

The simplified example of TLOG data is shown in Table 2 for the purposeof illustrating the solution of the constant of proportionality α_(ij)in affinity and cannibalization model 140. The first line item showsthat on day/time D1 (date and time) store S1 had transaction T1 in whichcustomer C1 purchased one product P1 at 1.50. In transaction T2 onday/time D1, customer C2 has one product P1 at price 1.50 each and oneproduct P2 at price 3.00. In transaction T3, customer C3 has one productP2 at 3.00. The remaining transactions T4-T10 on day/time D1 in store S1involved other products P3-P9 as shown.

The TLOG data in Table 2 further shows that products P1 in transactionsT1 and T2 had promotion PROMO1. Again, to simplify the explanation, noother product in Table 2 has any promotion.

The probability of having products P1 and P2 in the same transaction,assuming that the products are uncorrelated, is the number oftransactions containing product P1 over the total number of transactionstimes the number of transactions containing product P2 over the totalnumber of transactions. The uncorrelated base probability is ( 2/10)*(2/10)=0.04. The actual probability of observing the products together isthe number of transactions containing both products divided by the totalnumber of transactions is ( 1/10)=0.1>0.04. Accordingly, the presumptionis that these products are positively correlated, i.e., that there is anaffinity relationship between them.

The conditional probability Γ(P2|P1) is the number of basket withproducts P1 and P2 over total number of baskets divided by the number ofbasket with product P1 over total number of baskets. From the TLOG datain Table 2, Γ(P1|P2)=( 1/10)/( 2/10)=0.5. Γ(P2| P1 ) is total basketscontaining product P2 and no product P1 over total baskets divided bytotal baskets containing no product P1 over total number of baskets.From the TLOG data in Table 2, Γ(P2| P1 )=( 1/10)/( 8/10)=0.125. Fromequation (37), α_(P2P1) is Γ(P2|P1)−Γ(P2| P1 )=0.5−0.125=0.375. Sinceα_(P2P1) is greater than zero, the constant of proportionality α_(P2P1)defines an affinity relationship between products P1 and P2. If α_(ij)had been less than zero, then the products would have a cannibalizationrelationship. An offset threshold τ may be assigned to the constant ofproportionality such that an affinity relationship exists if α_(ij) isgreater than τ, and a cannibalization relationship exists if α_(ij) isless than −τ. In practice, the offsets for affinity and cannibalizationmay be asymmetric.

Returning to equation (32), the Δb_(P1)(D1) in equation (32) is thechange in b_(P1)(D1), usually an increase in basket share, associatedwith promotion PROMO1 of product P1 over time period D1, i.e.promotional lift. With the constant of proportionality α_(P2P1) knownfrom the TLOG data, the expectation of Δb_(P2)(D1) can be determined. IfΔb_(P1)(D1) changes by 10 due to promotion PROMO1, then Δb_(P2)(D1)changes by 10*0.375=3.75 due to the promotion PROMO1 on product P1.

The process of forecasting an affinity and cannibalization model usingobservable data from customer purchase decisions is shown in FIG. 11.Step 150 provides a linear relationship between functions related tofirst and second products. The linear relationship includes a constantof proportionality between the functions related to the first and secondproducts. The linear relationship between the functions related to thefirst and second products relates basket share of the first product tobasket share of the second product by the constant of proportionality,i.e. the linear relationship is given by Δb_(i)(t)=α_(ij)Δb_(j)(t),where α_(ij) is the constant of proportionality. Step 152 expresses thelinear relationship using a Taylor Series expansion. Step 154 expressesthe constant of proportionality using a first derivative term of theTaylor Series expansion. Step 156 solves for the constant ofproportionality using data observable from customer transactions. Thedata observable from customer transactions includes transaction,product, and promotion. In step 158, the constant of proportionalityrepresents an affinity relationship between the first and secondproducts if the constant of proportionality is greater than zero. Instep 160, the constant of proportionality represents a cannibalizationrelationship between the first and second products if the constant ofproportionality is less than zero.

The analysis of report 142, as generated by affinity and cannibalizationmodel 140, helps explain the effect of affinity and cannibalization oncustomer response in terms of unit sales, revenue, and profitability.Retailer 10 can use model 140 to forecast unit sales per product overtime, under a variety of promotional programs, and combinations ofpromotional programs. More specifically, retailer 10 can understand theeffect on unit sales of one product as influenced by a promotionaloffering on another product, where there is an affinity orcannibalization relationship between the two products. Understanding thecause and effect behind promotional offerings and inter-productrelationships is important to increasing the profitability of the retailstores. The promotional model addresses effective analysis techniquesfor various promotions, in terms of forecasting and backcasting, andprovides tools for a successful, scientific approach to promotionalprograms with a high degree of confidence. Although unit sales in theretail environment has been discussed in detail as the economic model,affinity and cannibalization model 140 is applicable to many othereconomic, scientific, and commercial decision processes.

While one or more embodiments of the present invention have beenillustrated in detail, the skilled artisan will appreciate thatmodifications and adaptations to those embodiments may be made withoutdeparting from the scope of the present invention as set forth in thefollowing claims.

1. A computer readable medium, comprising instructions that whenexecuted by a processor causes the computer to perform the stepsimplemented method of modeling customer response, comprising: providingin an affinity and cannibalization model a linear relationship between afirst basket share function b_(i)(t) describing transaction frequencyfor a first product P_(i) purchased and a second basket share functionb_(j)(t) describing transaction frequency for a second product P_(j)purchased as determined from transaction log data, wherein the linearrelationship includes a constant of proportionality between the firstand second functions estimated as the difference between a probabilityof the customer purchasing P_(i) given that P_(j) is already selectedfor purchase, and a probability of a customer purchasing P_(i) giventhat P_(j) is not selected for purchase, given by the followingequation: {acute over (α)}_(ij)=Γ(P_(i)|P_(j))−Γ(P_(i)|−P_(j)), where Γ() is the conditional probability on the first and second products P_(i)and P_(j), and {acute over (α)}_(ij) is the constant of proportionality;and solving for the constant of proportionality using the transactionlog data.
 2. The computer readable medium of claim 1, wherein thetransaction log data includes transaction, product, and promotion. 3.The computer readable medium of claim 1, wherein the linear relationshipbetween the functions b_(i)(t) and b_(j)(t) relates basket share of thefirst product to basket share of the second product by the constant ofproportionality.
 4. The computer readable medium of claim 1, whereinb_(i)(t) represents a number of transactions containing at least oneproduct P_(i) per time period t; and where b_(j)(t) represents a numberof transactions containing at least one product P_(j) per time period t.5. The computer readable medium of claim 1, wherein the linearrelationship is given by Δb_(i)(t)=α_(ij)Δb_(j)(t), where α_(ij) is theconstant of proportionality.
 6. The computer readable medium of claim 1wherein the affinity and cannibalization model approximates the constantof proportionality by a ratio of a rate of change of basket share forthe second product to a rate of change of basket share for the firstproduct.
 7. The computer readable medium of claim 1 wherein the firstproduct is offered under a promotion.
 8. The computer readable medium ofclaim 1 wherein the affinity and cannibalization model provides a baseprobability of observing the first and second products in a transaction,given the first and second products are uncorrelated.
 9. The computerreadable medium of claim 8, wherein the constant of proportionalityrepresents an affinity relationship between the first and secondproducts if the constant of proportionality is greater than zero. 10.The computer readable medium method of claim 9, wherein the affinityrelationship models an increase in basket share of the second productattributed to an increase in basket share of the first product.
 11. Thecomputer readable medium of claim 8, wherein the constant ofproportionality represents a cannibalization relationship between thefirst and second products if the constant of proportionality is lessthan zero.
 12. The computer readable medium of claim 11, wherein thecannibalization relationship models a decrease in basket share of thesecond product attributed to an increase in basket share of the firstproduct.
 13. A computer readable medium, comprising instructions thatwhen executed by a processor causes the computer to perform the stepsmethod of providing a computer model of customer response, comprising:providing in an affinity and cannibalization model a linear relationshipbetween a first basket share function b_(i)(t) describing transactionfrequency for a first product P_(i) purchased and a second basket sharefunction b_(j)(t) describing transaction frequency for a second productP_(j) purchased as determined from data observable from customerresponses, wherein the linear relationship includes a constant ofproportionality between the first and second functions estimated as adifference between a probability of a customer purchasing P_(i) giventhat P_(j) is already selected for purchase, and a probability of acustomer purchasing P_(i) given that P_(j) is not selected for purchase, given by the following equation: {acute over(α)}_(ij)=Γ(P_(i)|P_(j))ΓΓ(P_(i)|−P_(j)), where Γ( ) is the conditionalprobability on the first and second products P_(i) and P_(j), and {acuteover (α)}_(ij) is the constant of proportionality; and solving for theconstant of proportionality using data observable from customerresponses—the constant of proportionality representing an affinity orcannibalization relationship between the first and second products. 14.The computer readable medium of claim 13, wherein the linearrelationship between the functions relates basket share of the firstproduct to basket share of the second product by the constant ofproportionality.
 15. The computer readable medium of claim 13, whereinb_(i)(t) represents a number of transactions containing at least oneproduct P_(i) per time period t; and where b_(j)(t) represents a numberof transactions containing at least one product P_(j) per time period t.16. The computer readable medium of claim 13, wherein the linearrelationship is given by Δb_(i)(t)=α_(ij)Δb_(j)(t), where α_(ij) is theconstant of proportionality.
 17. The computer readable medium of claim13, wherein the affinity and cannibalization model provides a baseprobability of observing the first and second products in a transaction,given the first and second products are uncorrelated.
 18. The computerreadable medium of claim 17, wherein the constant of proportionalityrepresents the affinity relationship between the first and secondproducts if the constant of proportionality is greater than zero. 19.The computer readable medium of claim 17, wherein the constant ofproportionality represents the cannibalization relationship between thefirst and second products if the constant of proportionality is lessthan zero.
 20. A computer readable medium, comprising instructions thatwhen executed by a processor causes the computer to perform the steps,comprising: providing in an affinity and cannibalization model a linearrelationship between a first basket share function b_(i)(t) describingtransaction frequency for a first product P_(i) purchased and a secondbasket share function b_(j)(t) describing transaction frequency for asecond product P_(j) purchased as determined by data observable fromcustomer responses, wherein the model includes a constant ofproportionality between the first and second functions estimated as thedifference between a probability of the customer purchasing P_(i) giventhat P_(j) is already selected for purchase, and a probability of acustomer purchasing P_(i) given that P_(j) is not selected for purchase,given by the following equation: {acute over(α)}_(ij)=Γ(P_(i)|P_(j))−Γ(P_(i)|−P_(j)), where Γ( ) is the conditionalprobability on the first and second products P_(i) and P_(j), and {acuteover (α)}_(ij) is the constant of proportionality; and solving for theconstant of proportionality using the data observable from customerresponses, the constant of proportionality representing an affinity orcannibalization relationship between the first and second productspurchased.
 21. The computer readable medium of claim 20, wherein theaffinity and cannibalization model relates basket share of the firstproduct to basket share of the second product by the constant ofproportionality.
 22. The computer readable medium of claim 20, whereinb_(i)(t) represents a number of transactions containing at least onefirst product P_(i) per time period t; and where b_(j)(t) represents anumber of transactions containing at least one second product P_(j) pertime period t.
 23. The computer readable medium of claim 20, wherein theaffinity and cannibalization model is given byΔb_(i)(t)=α_(ij)Δb_(j)(t), where α_(ij) is the constant ofproportionality.
 24. The computer readable medium of claim 20, whereinthe affinity and cannibalization model provides a base probability ofobserving the first and second products in a transaction, given thefirst and second products are uncorrelated.
 25. The computer readablemedium of claim 24, wherein the constant of proportionality representsthe affinity relationship between the first and second products if theconstant of proportionality is greater than zero.
 26. The computerreadable medium of claim 24, wherein the constant of proportionalityrepresents the cannibalization relationship between the first and secondproducts if the constant of proportionality is less than zero.
 27. Acomputer readable medium, comprising instructions that when executed bya processor causes the computer to perform the steps, comprising:relating a first basket share function b_(i)(t) describing transactionfrequency for a first product P_(i) purchased and a second basket sharefunction b_(j)(t) describing transaction frequency for a second productP_(j) purchased, by a constant of proportionality; estimating theconstant of proportionality as a difference between a probability of thecustomer purchasing P_(i) given that P_(j) is already selected forpurchase, and a probability of a customer purchasing P_(i) given thatP_(j) is not selected for purchase, given by the following equation:{acute over (α)}_(ij)=Γ(P_(i)|P_(j))−Γ(P_(i)|−P_(j)), where Γ( ) is theconditional probability on the first and second products P_(i) andP_(j), and {acute over (α)}_(ij) is the constant of proportionality; andsolving for the constant of proportionality using data observable fromcustomer responses, wherein the constant of proportionality representsaffinity or cannibalization relationship between the first and secondproducts purchased.
 28. The computer readable medium of claim 27,wherein the computer model relates basket share of the first product tobasket share of the second product by the constant of proportionality.29. The computer readable medium of claim 27, further includingproviding a base probability of observing the first and second productsin a transaction, given the first and second products are uncorrelated.30. The computer readable medium of claim 29, wherein the constant ofproportionality represents the affinity relationship between the firstand second products if the constant of proportionality is greater thanzero.
 31. The computer readable medium of claim 29, wherein the constantof proportionality represents the cannibalization relationship betweenthe first and second products if the constant of proportionality is lessthan zero.